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    <h4>Network inference</h4>
    <div class="main-paragraph">
        Deriving models of a (biological) system is called
        <span class="stressed-text">model inference</span>.<br>
        The objective of model inference is to find a set of parameters such that the model equations:
        <ul>
            <li>Best reproduce (have low error) an experimental training data (perturbation response data)</li>
            <li>Have predictive power beyond the training data (predict response to untested perturbations)</li>
        </ul>
    </div>
    <div class="main-image">
        <img src="images/network_inference.png" class="img-responsive" alt="BP mel">
    </div>
    <h4>Network inference is a hard problem</h4>
    <div class="main-paragraph">
        Computation of the cost of all possible network configurations will,
        in principle, lead to inference of optimal network configurations.
        However, explicit enumeration and cost calculation of all possible
        parameter configurations is a prohibitively complicated task for even
        moderately sized systems. To circumvent this problem, we have adapted
        from statistical physics a two-step approach. The approach is based on
        first calculating probability distributions for each possible interaction
        with Belief propagation algorithm, and then computing distinct solutions
        by sampling the probability distributions.
    </div>
    <h4>What is belief propagation?</h4>
    <div class="main-paragraph">
        Belief propagation (BP) is a message passing algorithm for probabilistic
        inference on graphical models. BP exploits a mean-field like cavity
        approach, in which the probability distribution of the variables in a
        <span class="stressed-text">cavity</span> can be defined by
        collective statistical properties of
        the surrounding variables. Thus, the distribution of the parameters
        can be treated by statistical mechanics principles; the probability
        distribution for each parameter value is defined by the Boltzmann
        distribution and computed numerically. BP works by passing mathematical
        messages on a factor graph between model parameters and experimental
        constraints iteratively until consecutive messages converge.
        Explicitly, the message passing equations are iteratively calculated
        until convergence. The result of BP is a set of probability distributions
        for all unfixed model parameters. By means of the probabilistic approach,
        the time-complexity of the problem is reduced and the obstacles imposed
        by combinatorial complexity are circumvented. Next, thousands of distinct
        networks models are instantiated from probability distributions with
        BP-guided decimation algorithm.
    </div>
    <div class="main-image">
        <img src="images/bp_algorithm.png" class="img-responsive" alt="BP mel">
    </div>
    <h4>Iteration process for Belief Propagation</h4>
    <div class="main-paragraph">
        Top panel: the global information consists of collecting the probability
        distributions of the non-cavity parameters without the contribution from
        the cavity condition. This is a simple product over all
        &rho;<sup>v</sup>(w<sub>ij</sub>)
        factors except that from the cavity constraint &mu;.
        Distributions centered on zero denote unlikely interactions (see j = 2),
        centered on the right of zero denote likely positive interactions (see j = 3),
        and centered on the left denote likely negative interactions (see j = N).
        These distributions inform the parameters of the Gaussian distribution
        for the mean-field, aggregate sum variable s<sup>&mu;</sup><sub>k</sub>.
        The distribution P<sup>&mu;</sup>(s<sup>&mu;</sup><sub>k</sub>)
        summarizes the state of the non-cavity parameters.<br>
        <br>
        Bottom panel: we calculate the probability of each possible parameter assignment
        &omega;&isin;&Omega;
        to the cavity parameter w<sub>ik</sub>
        constrained to the data in the cavity condition. This calculation boils down
        to a simple convolution of the fitness function with a fixed parameter assignment
        F<sup>&mu;</sup>(s<sup>&mu;</sup><sub>k</sub>)
        with the probability of the aggregate sum variable
        P<sup>&mu;</sup>(s<sup>&mu;</sup><sub>k</sub>),
        obtained by integrating over all values of
        s<sup>&mu;</sup><sub>k</sub>.
        Each assignment
        &omega;&isin;&Omega;
        contributes proportional to the area under the curve.
        The resulting update is the contribution of condition &mu;
        on the distribution of w<sub>ik</sub>,
        denoted &rho;<sup>&mu;</sup>(w<sub>ik</sub>).
        This recently updated distribution becomes part of
        the global information for successive updates to other parameters.
    </div>
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